$SL_k$-Tilings and Paths in $\mathbb{Z}^k$

TL;DR

The paper generalizes Short’s correspondence between SL_2-tilings and Farey-paths to higher rank by introducing bi-infinite $k$-vectors paths and a determinant-based map $m_{i j}=\det(\gamma_i,\ldots,\gamma_{i+k-2},\delta_j)$. It proves a bijection between tame $SL_k$-tilings and pairs of paths modulo $SL_k(\mathbb{Z})$, leveraging Plücker friezes and Grassmannian cluster algebras to connect to positivity, periodicity, and duality. It then derives consequences for periodic tilings, the relationship with $SL_k$-friezes, and a duality framework that mirrors classical results for $k=2$, while providing new positivity results for small $n$ and $k$ via Plücker coordinates and Gale duality. Overall, the work extends the Farey-graph–tiling paradigm to higher dimensions, offering a robust algebraic toolkit for understanding $SL_k$-tilings through Grassmannian and cluster-algebra structures with potential implications for positivity and combinatorial geometry.

Abstract

An $SL_k$-tiling is a bi-infinite array of integers having all adjacent $k\times k$ minors equal to one and all adjacent $(k+1)\times (k+1)$ minors equal to zero. Introduced and studied by Bergeron and Reutenauer, $SL_k$-tilings generalize the notion of Conway-Coxeter frieze patterns in the case $k=2$. In a recent paper, Short showed a bijection between bi-infinite paths of reduced rationals in the Farey graph and $SL_2$-tilings. We extend this result to higher $k$ by constructing a bijection between $SL_k$-tilings and certain pairs of bi-infinite strips of vectors in $\mathbb{Z}^k$ called paths. The key ingredient in the proof is the connection to Plücker friezes and Grassmannian cluster algebras. As an application, we obtain results about periodicity, duality, and positivity for tilings.

Figures (3)

• Figure 1: An $SL_2$-frieze of width $2$. Every $2\times 2$ diamond must have determinant $1$, for example $a1bc=ac-b=1$.
• Figure 2: The Plücker frieze $\mathcal{F}_{(2,5)}$ of type $(2,5)$. When applied to a matrix satisfying the assumptions in Theorem \ref{['thm:pluckerfrieze']}, the consecutive Plücker coordinates in rows $2$ and $5$ become all $1$'s and the remaining entries become integers, resulting in a frieze as in Figure \ref{['friezeexample']}.
• Figure 3: Tiling resulting from rotating and extending the frieze in Figure \ref{['friezeexample']}.

Theorems & Definitions (124)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• Remark 2.6
• Definition 2.7